In the Wikipedia article "Quantum Gravity", it claims that there is some experimental motivation for finding a quantum theory of gravity. In one of the experiments it cites, neutrons are found to jump between discrete quantum states in the earth's gravitational potential, similar to the discrete states of an electron in the nucleus's electrostatic potential.

I don't know much about quantum gravity, but it seems that from the equivalence principle, we could view the above experiment as simply what would happen to a neutron in an accelerated reference frame of 9.8 m/s^2. In fact, couldn't one replicate this experiment in a spaceship accelerating at 9.8 m/s^2 and get the same result, without any force of gravity? So there seems to be no evidence for the force of gravity in this experiment, let alone quantum gravity.

In the Wikipedia article "Quantum Gravity", it claims that there is some experimental motivation for finding a quantum theory of gravity. In one of the experiments it cites, neutrons are found to jump between discrete quantum states in the earth's gravitational potential, similar to the discrete states of an electron in the nucleus's electrostatic potential.

There's nothing wrong with their interpretation. It's just not a very exciting interpretation. Quantum gravity is not a total mystery. We don't need a complete theory of quantum gravity to calculate certain things about quantum gravity. This is one of those things we can calculate.

I don't know much about quantum gravity, but it seems that from the equivalence principle, we could view the above experiment as simply what would happen to a neutron in an accelerated reference frame of 9.8 m/s^2. In fact, couldn't one replicate this experiment in a spaceship accelerating at 9.8 m/s^2 and get the same result, without any force of gravity? So there seems to be no evidence for the force of gravity in this experiment, let alone quantum gravity.

There is no meaningful distinction between "real" gravity and what we experience in an accelerated frame, so it doesn't make sense to worry about which one the experiment detects.

But there is a distinction between real gravity and an accelerated reference frame. Real gravity involves tidal forces which can be represented as the curvature of space-time.

Also, even if this experiment involved tidal forces, I'm still not sure it would be evidence for a quantum theory of gravity. An analogy is that the discrete states of electrons in atoms are not evidence for quantum electrodynamics. Rather, they are the result of an approximation in which the classical electrostatic potential is used in the Schrodinger equation.

But there is a distinction between real gravity and an accelerated reference frame. Real gravity involves tidal forces which can be represented as the curvature of space-time.

Good point -- it's true that that experiment didn't test anything related to curvature. However, it's not true that you don't get nontrivial quantum gravity effects without curvature. For example, it's believed that you would see Hawking radiation from the event horizon of an accelerated observer in flat spacetime. BTW, tidal forces are not synonymous with curvature. Tidal forces are one type of curvature. The type of curvature measured by the Einstein tensor is non-tidal.

I guess what I'm asking is this:
Is it true that if you do this experiment in an accelerating spaceship you would get the same discrete states?

And if the above is true, then you don't need a quantum theory of gravity to explain the results of the experiment. All you need to do is a quantum theory of the neutron in an accelerated reference frame.

If the equivalence principle holds, then I imagine the same results would hold in either case. I don't imagine the neutrons experience tidal forces and the whole idea of GR is that in small enough regions of spacetime we can't tell gravity apart from acceleration. In larger regions we can though.

I don't see this as evidence for the existence of quantum gravity. It is nice to see that gravity can play a role in quantum mechanics, but the quantum nature of gravity isn't explored. That being said, I could see how such an experiment could serve as a good thought experiment, but I imagine the required accuracy to test ideas of quantum gravity would be forbiddingly high.

I don't think this is correct. At least if we agree that Newtonian tidal forces correspond to Riemannian curvature in GR.
It is true tidal forces are usually associated with the Weyl tensor in vacuum solutions with vanishing Einstein tensor because that is the only part that is left of the Riemannian curvature in vacuum.
In the non-vacuum case there is a stress-energy tensor source so there is Einstein tensor curvature manifested as tidal forces. At least that is what the equation Gab=8piGTab seems to indicate.